Multiproduct-Firm Oligopoly: An Aggregative Games Approach

Transcription

1 Multiproduct-Firm Oligopoly: An Aggregative Games Approach Volker Nocke Nicolas Schutz September 21, 2016 Abstract We develop an aggregative games approach to study oligopolistic price competition with multiproduct firms. We introduce a new class of demand systems, derived from discrete/continuous choice, and nesting CES and logit demand systems. The associated pricing game with multiproduct firms is aggregative and a firm s optimal price vector can be summarized by a uni-dimensional sufficient statistic, the ι-markup. We prove existence of equilibrium using a nested fixed-point argument, and provide conditions for equilibrium uniqueness. In equilibrium, firms may choose not to offer some products. We analyze the pricing distortions and provide monotone comparative statics. Under CES and logit demands, another aggregation property obtains: All relevant information for determining a firm s performance and competitive impact is contained in that firm s uni-dimensional type. Finally, we re-visit classic questions in static and dynamic merger analysis, and study the impact of a trade liberalization on the inter- and intra-firm size distributions, productivity and welfare. 1 Introduction Analyzing the behavior of multiproduct firms in oligopolistic markets appears to be of a firstorder importance. Multiproduct firms are endemic and play an important role in the economy. Even when defining products quite broadly at the NAICS 5-digit level, multiproduct firms account for 91% of total output and 41% of the total number of firms Bernard, Redding, We thank John Asker, olger Breinlich, Martin Jensen, Andreas Kleiner, Tim Lee, Patrick Rey, Michael Riordan, Glen Weyl, as well as seminar and conference participants at Columbia-Duke-MIT-Northwestern IO Theory Workshop 2015, EARIE 2016, ESAM 2016, MaCCI Annual Meeting 2016, MaCCI Summer Institute 2015, MaCCI Workshop on Multiproduct Firms, SEARLE Antitrust Conference 2015, SFB TR/ , Copenhagen BS, U Berlin, IESE, Kyoto, Lausanne, Leicester, LSE, MIT/arvard, Nagoya, Pompeu Fabra, UCLA and Vienna for helpful comments and suggestions. The first author gratefully acknowledges financial support from the European Research Council ERC grant no ). UCLA, University of Mannheim, and CEPR. volker.nocke@gmail.com. University of Mannheim. schutz@uni-mannheim.de. 1

2 and Schott, 2010). Similarly, many markets are characterized by oligopolistic competition. Even at the 5-digit industry level, concentration ratios are fairly high: For instance, in U.S. manufacturing, the average NAICS 5-digit industry has a four-firm concentration ratio of 35% Source: Census of U.S. Manufacturing, 2002). 1 The ubiquitousness of multiproduct firms and oligopoly is reflected in the modern empirical IO literature, where oligopolistic price competition with multiproduct firms abounds e.g., Berry, 1994; Berry, Levinsohn, and Pakes, 1995; Nevo, 2001). In contrast to single-product firms, a multiproduct firm must choose not only how aggressive it wants to be in the market place but also how to vary its markups across products within its portfolio. In contrast to monopolistically competitive firms, an oligopolistic multiproduct firm must take self-cannibalization into account, both when setting its markups and when deciding which products to offer. Multiproduct-firm oligopoly therefore gives rise to a number of important questions: What determines the within-firm markup structure, between-firm markup differences, and the industry-wide markup level? What explains firms scope in oligopoly? Along which dimensions are markups and product offerings distorted by oligopolistic behavior? Due to the technical difficulties discussed below, these questions have been under-researched in the existing literature. In this paper, we develop an aggregative games approach to circumvent the technical difficulties and address these and related questions. We make several contributions. We introduce a new class of integrable) quasi-linear demand systems, derived from a discrete/continuous choice model of consumer demand, where each consumer first chooses which product to purchase, and then how much of that product to consume. This class nests standard constant elasticity of substitution CES) and multinomial logit MNL) demand systems as special cases. As demand satisfies the Independence of Irrelevant Alternatives IIA) axiom, consumer surplus depends only on an aggregator that is additively separable in prices. We use this class of demand systems to analyze oligopolistic price competition between multiproduct firms with arbitrary firm and product heterogeneity. The associated pricing game has two important properties. First, it is aggregative, and the aggregator is the same as the one for consumer surplus. Second, a firm s optimal price vector is such that, for every product in that firm s portfolio, the Lerner index multiplied by a product-level elasticity measure is equal to a firm-level sufficient statistic, called the ι-markup. That ι-markup pins down the price level of the firm. 2 We can therefore think of the firm s maximization problem as one of choosing the right ι-markup. These two properties allow us to prove existence of a pricing equilibrium under weak con- 1 The prevalence of multiproduct-firms and oligopolistic markets would be larger if one were to define products and industries more narrowly, as is usually done in industrial organization. The focus of this paper will be on such narrowly defined industries in which there are substantial demand-side linkages between all products. 2 With CES resp. MNL) demands, this implies that the firm charges the same relative resp. absolute) markup over all its products. Our class of demand systems allows us to generate heterogeneous relative and absolute markups within a firm s set of products. 2

3 ditions using a nested fixed point argument. This approach circumvents problems that arise when attempting to apply off-the-shelf equilibrium existence theorems, such as the failure of quasi-concavity, log-)supermodularity and upper semi-continuity of the profit functions. It also gives rise to an efficient algorithm for computing equilibrium, and allows us to derive sufficient conditions for equilibrium uniqueness. The aggregative structure of the game and the constant ι-markup property allow us to decompose the welfare distortions from oligopolistic competition between multiproduct firms as follows. First, since firms are setting positive markups, the industry delivers too little consumer surplus. Second, consumer surplus is not delivered efficiently: Although a social planner would like to set the same ι-markup on all the products in the industry, equilibrium ι-markups typically differ across firms. This means that some firms are inefficiently large, while others are inefficiently small. Perhaps surprisingly, there are no within-firm pricing distortions, in the sense that a firm s equilibrium prices maximize social welfare subject to the constraint that that firm s contribution to consumer surplus is held fixed at its equilibrium value. We also study the determinants of firm scope. Despite the absence of product-specific fixed costs, firms do not necessarily sell all their products in equilibrium. The intuition is that a multiproduct oligopolist has to worry about self-cannibalization effects when setting its prices and when deciding which products to supply. It therefore has incentives to withdraw some of its weaker products so as to channel consumers toward its more profitable products. Despite our game not being supermodular, we are able to rank equilibria from the consumers and firms viewpoints, and to perform comparative statics on the set of equilibria. We explore the impact of entry, trade liberalization, and productivity and quality shocks on industry conduct and performance. Among other results, we find that a shock that makes the industry more competitive such as a trade liberalization or the entry of new competitors) induces firms to broaden their scope in equilibrium. This is in stark contrast with existing results in the international trade literature on multiproduct firms, which we further discuss in Section 1.1. Intuitively, as the industry becomes more competitive, a firm starts worrying more about consumers purchasing its rivals products, and less about cannibalizing its own sales. That firm therefore has incentives to introduce fighting brands Johnson and Myatt, 2003) to protect its market share. In the last part of the paper, we specialize the model to the cases of CES and MNL demands. We show that an additional aggregation property, called type aggregation, obtains: All relevant information for determining a firm s performance and competitive impact is contained in that firm s uni-dimensional type. This property allows us to obtain additional predictions e.g., about the impact of productivity and quality shocks on social welfare) that are unavailable in the general case. We provide two sets of applications of our framework with CES and MNL demands. First, we revisit classical questions in static and dynamic merger analysis. These issues have been addressed in the literature using the single-product homogeneous-goods Cournot oligopoly model, which is a well-known example of an aggregative game see Farrell and Shapiro, 1990; Nocke and Whinston, 2010). The aggregative 3

4 structure of our game and the type aggregation property allow us to generalize the insights of that earlier literature to settings with multiproduct firms, price competition and horizontally differentiated products. Second, we study the impact of a unilateral trade liberalization on the inter- and intra-firm size distributions, average industry-level productivity, and welfare. Road map. In the remainder of this section, we provide a short review of the related literature. In Section 2, we analyze discrete/continuous consumer choice and derive a new class of demand systems. In Section 3, we describe the multiproduct-firm pricing game. This is followed, in Section 4, by the equilibrium analysis. We prove existence of equilibrium under mild conditions and uniqueness of equilibrium under stronger conditions. We characterize the equilibrium pricing structure as well as firms scope, provide a welfare analysis, and perform monotone comparative statics. In Section 5, we specialize to the cases of CES and MNL demands and show that the type aggregation property obtains. In Section 6, we study applications to static and dynamic merger analysis and to the impact of international trade. Finally, we conclude in Section Related Literature Our paper contributes to the relatively small literature on multiproduct-firm oligopoly pricing with horizontally differentiated products. One strand of that literature focuses on proving equilibrium existence and uniqueness in multiproduct-firm oligopoly pricing games with firm and product heterogeneity and demand systems derived from discrete/continuous choice. Importantly, Caplin and Nalebuff 1991) s powerful existence theorem for pricing games with single-product firms, the proof of which relies on establishing quasi-concavity of a firm s profit function in own price, does not extend to the case of multiproduct firms. The reason is that, even with standard MNL demand, a multiproduct firm s profit function often fails to be quasi-concave Spady, 1984; anson and Martin, 1996). For this reason, the literature has focused on special cases of discrete/continuous choice demand systems, such as MNL demand Spady, 1984; Konovalov and Sándor, 2010), CES demand Konovalov and Sándor, 2010), and nested logit demand where each firm owns a nest of products Gallego and Wang, 2014). Our aggregative games techniques provide a unified approach to address existence and uniqueness issues in pricing games with discrete/continuous choice demand systems. More applied papers make stronger functional forms and/or symmetry assumptions, and study firms product range decisions in oligopoly. Anderson and de Palma 1992, 2006) analyze the equilibrium resp. welfare-maximizing) number of firms and number of products per firm in a model with discrete/continuous choice demands, symmetric products and firms, and free entry. Shaked and Sutton 1990) and Dobson and Waterson 1996) develop lineardemand models to analyze product-range decisions. The tools developed in the present paper allow us to do away with the symmetry and linearity assumptions made in these earlier 4

5 papers. 3 More recently, multiproduct firms have received much attention from international trade researchers. A central question in that literature is how multiproduct firms react to trade liberalization by adjusting their product range and product mix. This question is usually addressed in models of monopolistic competition, under either CES demand Bernard, Redding, and Schott, 2010, 2011; Nocke and Yeaple, 2014) or linear demand Dhingra, 2013; Mayer, Melitz, and Ottaviano, 2014). An exception is Eckel and Neary 2010) who study identical) multiproduct firms in a Cournot model with linear demand. 4 A common finding in these papers is that firms react to a trade liberalization by refocusing on their core competencies, i.e., by shrinking their product ranges. 5 In models with CES demand and product-specific fixed costs, this is due to the fact that more intense competition reduces variable profits on all products, and therefore makes it harder to cover fixed costs. In models with linear demand, more intense competition chokes out the demand for products sold at a high price. By contrast, we find that self-cannibalization effects matter relatively less when competition is more intense, which implies that firms respond to trade liberalization by broadening their product ranges. The concept of aggregative games was introduced by Selten 1970). In such games, a player s payoff depends only on that player s actions and on an aggregate common to all players. In games with additive aggregation, each player has a fitting-in correspondence, and the set of pure-strategy Nash equilibria corresponds to the set of fixed points of the aggregate fitting-in correspondence. 6 McManus 1962, 1964) and Selten 1970) use the aggregate fitting-in correspondence to establish existence of a Nash equilibrium in a homogeneous-goods Cournot model. Szidarovszky and Yakowitz 1977), Novshek 1985) and Kukushkin 1994) refine this approach further. 7 Our proof of equilibrium existence and our characterization of the set of equilibria also rely on the aggregate fitting-in correspondence. Corchon 1994) and Acemoglu and Jensen 2013) show that aggregative games also deliver powerful monotone comparative statics results, in the spirit of Milgrom and Roberts 1994) and Milgrom and Shannon 1994). We perform such monotone comparative statics in Section 4.3. In a recent paper, Anderson, Erkal, and Piccinin 2013) adopt an aggregative games approach to study pricing games similar to ours, but restrict attention to single-product firms. They are mainly interested in long-run comparative statics with free entry and exit. 3 Another strand of literature studies price and quantity competition between multiproduct firms selling vertically differentiated products. See, among others, Champsaur and Rochet 1989), and Johnson and Myatt 2003, 2006). 4 Eckel, Iacovone, Javorcik, and Neary 2015) extend their framework to heterogeneous firms. 5 In Nocke and Yeaple 2014), the prediction depends on whether the firm sells only domestically or not. 6 See Section 4.1 for more details. The term fitting-in correspondence was coined by Selten 1973). This concept appears under different names in the literature. 7 The aggregate fitting-in correspondence is not well-defined when aggregation is not additive Cornes and artley, 2012). Dubey, aimanko, and Zapechelnyuk 2006) and Jensen 2010) show that games with nonadditive aggregation and monotone best replies have a pseudo potential or a best-reply potential depending on the notion of monotonicity employed). This allows them to establish equilibrium existence. In Section 4.4, we show that our multiproduct-firm pricing game has an ordinal potential Monderer and Shapley, 1996). 5

6 The fact that a firm optimally sets the same absolute markup possibly adjusted by a price-sensitivity parameter) over all its products when demand is of the MNL type, and the same relative markup over all its products when demand is of the CES type, was already pointed out by Anderson, de Palma, and Thisse 1992), Konovalov and Sándor 2010), and Gallego and Wang 2014). The common ι-markup property, discussed above, generalizes these findings to the whole class of demand systems that can be derived from discrete/continuous choice, and allows us to simplify the firm s pricing problem considerably. Similar in spirit, Armstrong and Vickers 2016) reduce the dimensionality of a multiproduct monopolist s quantity-setting problem by confining attention to demand systems that have the property that consumer surplus is homothetic in quantities. They show that the multiproduct monopolist optimally scales down the welfare-maximizing vector of quantities by a common multiplicative factor. 8 2 Discrete/Continuous Consumer Choice We consider a demand model in which consumers make discrete/continuous choices: Each consumer first decides which product to purchase, and then, how much of this product to consume. This approach captures Novshek and Sonnenschein 1979) s idea that price-induced demand changes can be decomposed into two effects: An intensive margin effect consumers purchase less of the product whose price was raised), and an extensive margin effect some consumers stop purchasing the commodity whose price increased). 9,10 Discrete/continuous choice models of consumer demand have been used by empirical researchers to estimate demand for electric appliances Dubin and McFadden, 1984), soft drinks Chan, 2006), and painkillers Björnerstedt and Verboven, 2016). In Smith 2004), consumers first choose a supermarket, and then how much to spend at that supermarket based on the price index at that store. In addition to giving rise to tractable multiproduct-firm pricing games, the discrete/continuous approach will also turn out to be useful to interpret some of the comparative statics results derived in Section 4. We formalize discrete/continuous choice as follows. There is a population of consumers with quasi-linear preferences. Each consumer chooses a single product from a finite and nonempty set of products N and the quantity of that product; he spends the rest of his income on the outside good or icksian composite commodity), the price of which is normalized to one. Conditional on selecting product i, the consumer receives indirect utility y +v i p i )+ε i, where p i is the price of product i, y is the consumer s income, and ε i is a taste shock. By Roy s identity, the consumer purchases v ip i ) units of good i. We call v ip i ) the conditional 8 Their class of demand systems does not nest ours e.g., it does no include CES-like demand with heterogeneous price elasticity parameters) nor does ours nest theirs e.g., ours does not include linear demand). Armstrong and Vickers 2016) also extend Bergstrom and Varian 1985) to establish equilibrium existence in a Cournot oligopoly model with identical multiproduct firms. 9 Income effects are absent in our quasi-linear world. 10 See also anemann 1984). 6

7 demand for product i. At the product-choice stage, the consumer selects product i only if j N, y + v k p k ) + ε k y + v j p j ) + ε j. We assume that the components of vector ε j ) j N are identically and independently drawn from a type-1 extreme value distribution. By olman and Marley s theorem, product i is therefore chosen with probability ) P i p) = Pr v i p i ) + ε i = max v jp j ) + ε j ) = j N e v ip i ) = h i p i ) j N ev jp j ) j N h jp j ), where h j e v j for every j. It follows that expected demand for product k is given by P k p)q k p k ) = e v kp k ) j N ev jp j ) v kp k )) = h k p k) j N h jp j ). In the following, we use collection of functions h j ) j N rather than v j ) j N ) as primitives. We assume that all the h functions are C 3 from R ++ to R ++, strictly decreasing, and logconvex. The assumption that h j is non-increasing and log-convex is necessary and sufficient for v j to be an indirect subutility function Nocke and Schutz, 2016b). The assumption that h j is strictly decreasing means that the demand for product j never vanishes. To sum up, the demand system generated by the discrete/continuous choice model h j ) j N when normalizing market size to one) is: D k p j ) j N ) = h k p k) j N h jp j ), k N, p j) j N R N ++. 1) The conditional demand for good k is d log h k /dp k = h k /h k. Product k is chosen with probability h k / j h j. Our class of demand systems nests standard MNL if h j p j ) = e a j p j λ for all j N, where a j R and λ > 0 are parameters) and CES demands if h j p j ) = a j p 1 σ j, where a j > 0 and σ > 1 are parameters) as special cases. The fact that CES demands can be derived from discrete/continuous choice was already pointed out by Anderson, de Palma, and Thisse 1987) in a slightly different framework without an outside good. As seen in equation 1), the class of demand systems that are derivable from discrete/continuous choice is much wider than CES and MNL demands. Notice also that, by virtue of the i.i.d. type-1 extreme value distribution assumption, all these demand systems have the IIA property. That property will allow us to greatly simplify the multiproduct-firm pricing problem in Section 4. The consumer s expected utility can be computed using standard formulas see, e.g., 7

8 Anderson, de Palma, and Thisse, 1992): ) E y + max v jp j ) = y + log j N j N e v jp j ) ) ) = y + log h j p j ). 2) Therefore, consumer surplus is aggregative, in that it only depends on the value of the aggregator j N h jp j ). Representative consumer approach. While much of the empirical industrial organization literature has adopted discrete choice models as a way of deriving consumer demand, other strands of literature, such as the international trade literature, mainly use a representative consumer approach. 11 We show that demand system 1) is quasi-linearly integrable, i.e., it can be obtained from the maximization of the utility function of a representative consumer with quasi-linear preferences: 12 Proposition 1. Let D be the demand system generated by discrete/continuous choice model h j ) j N. D is quasi-linearly integrable. Moreover, v is an indirect subutility function for D if and only if there exists a constant α R such that vp j ) j N ) = α + log j N h jp j ). Proof. See Online Appendix I. ence, any demand system that can be derived from discrete/continuous choice can also be derived from quasi-linear utility maximization. The second part of the proposition says that the expected utility of a consumer engaging in discrete/continuous choice and the indirect utility of the associated representative consumer coincide up to an additive constant,). The results we will derive on consumer welfare therefore do not depend on the way the demand system has been generated. Whether we use discrete/continuous choice or a representative consumer approach, all that matters is the value of aggregator. Consumer heterogeneity. While the discrete/continuous consumer choice model allows for some type of consumer heterogeneity different consumers receive different taste shocks and may therefore select different products), it does have the property that all consumers who select the same product choose to purchase the same quantity. owever, the model can easily be adapted to accommodate consumer heterogeneity in the quantity purchased of the same 11 See however Fajgelbaum, Grossman, and elpman 2011), andbury 2013) and Fally and Faber 2016) for recent examples of international trade papers deriving demand from special cases of) discrete/continuous choice. 12 Quasi-linear integrability and indirect subutility functions are defined in Nocke and Schutz 2016b), Definitions 3 and 4. In Online Appendix I, we prove a more general result: We derive necessary and sufficient conditions for demand system j N ) g k p k ) D k p j ) j N = j N h jp j ), k N, p j) j N R N ++ to be quasi-linearly integrable. 8

9 product. In particular, suppose that the indirect subutility derived from choosing product j is v j p j, t j ), where t j R is the consumer s type for product j, drawn from probability distribution G j ). The realized value of t j is observed by the consumer only after he has chosen product j. Let v j p j ) = v j p j, t j )dg j t j ) be the expected indirect utility derived from product j. Then, product i is chosen with probability exp v i p i )/ j exp v jp j )). Under some technical conditions which allow us to differentiate under the integral sign), the consumer s expected conditional demand for product j is: v j p j, t j )dg j t j ) = v j p j, t j )dg j t j ) = v p j p jp j ). j Therefore, if we define h j p j ) = expv j p j )) for every j, then the expected unconditional) demand for product i is still given by equation 1). Differentiating once more under the integral sign, we also see that v j ) is decreasing and convex if v j, t j ) is decreasing and convex for every t j. Therefore, discrete/continuous choice with consumer heterogeneity gives rise to the same class of demand systems as discrete/continuous choice without heterogeneity The Multiproduct-Firm Pricing Game In this section, we describe the multiproduct-firm pricing game. The pricing game consists of three elements: h j ) j N is the demand system defined in the previous section; F, the set of firms, is a partition of N such that F 2; 14 c j ) j N R N ++ is a profile of marginal costs. In the following, we adopt the discrete/continuous choice interpretation of demand system h j ) j N but the reader should keep in mind that h j ) j N may equally well have been generated by utility maximization of a representative consumer.) The profit of firm f F is defined as follows: 15 Π f p) = k f p k < p k c k ) j N p j < h k p k) h j p j ) + j N p j = lim h j, p 0, ] N. 3) Note that we are allowing firms to set infinite prices, which essentially compactifies firms action sets. We will later see that this compactification ensures that each firm s maximization problem has a solution provided that rival firms are not pricing all their products at infinity). The assumption we are making here is that, if p k =, then the firm simply does not supply product k, and therefore does not earn any profit on this product. In the following, we write 13 If the consumer observes his vector of types before choosing a variety, then the implied demand system becomes a mixture of equation 1). We are not able to handle such mixtures of demand systems, because they no longer give rise to an aggregative game. This implies in particular that our approach cannot accommodate random coefficient logit demand systems. 14 That is, each product in N is offered by exactly one firm. 15 Throughout the paper, we adopt the convention that the sum of an empty collection of real numbers is equal to zero. Note that, since h j is monotone, lim h j exists for every j. 9

10 h j ) instead of lim h j. We say that product j is active if p j <. We discuss infinite prices in greater detail at the end of this section. We study the normal-form game in which firms set their prices simultaneously, and payoff functions are given by equation 3). A pure-strategy Nash equilibrium of that normal-form game is called a pricing equilibrium. Outside option. Since f F f = N, our definition of a pricing game does not seem to allow for an exogenously priced outside option. 16 owever, such an outside option is easy to incorporate. Let h j ) j N be a discrete/continuous choice model of consumer demand. Partition N into two sets: Ñ, the set of products sold by oligopoly players, and N 0, the set of products sold at exogenous prices p j ) j N 0. Let 0 = j N h 0 j p j ) > 0 be the value of the outside option. We can now define another discrete/continuous choice model h j ) j Ñ as follows: For every j Ñ, h j = h j + 0 Note that this transformation affects Ñ.17 neither consumer surplus nor expected demand. Therefore, the price competition game with discrete/continuous choice model h j ) j N and exogenous prices p j ) j N 0 is equivalent to the price competition game with discrete/continuous choice model h j ) j Ñ and no outside option. More on infinite prices. We first argue that the idea that product k is simply not supplied when p k = is consistent with the discrete/continuous choice interpretation of the demand system. In the discrete/continuous choice model, a consumer receives a type-1 extreme value draw ε k for product k even when p k =. Three cases can arise when the price is infinite: i) The conditional demand is positive lim h k /h k > 0), in which case the choice probability must be equal to zero lim h k = 0). ii) The choice probability is positive lim h k > 0), in which case the conditional demand must be equal to zero lim h k /h k = 0). iii) Both the conditional demand and the choice probability are equal to zero. 18 The term j:p j = lim h j appears in the denominator of Π f to allow for case ii). In all three cases, the consumer does not consume a positive quantity of the good when the price is infinite, which is consistent with the interpretation that the product is simply not available. An alternative way of allowing for infinite prices would be to define the profit function for finite prices first, and then extend it by continuity to price vectors that have infinite components. In the proof of Lemma B in the Appendix, we show that, if price vector p 0, ] N has a least one finite component, then lim p Π f coincides with the value of Π f p) defined in equation 3). There is, however, an important exception. If p j = for every j, 16 An exogenously priced outside option should not be confused with the outside good. For example, in the market for automobiles, public transport may represent the outside option whereas the outside good represents the expenditure on other goods. 17 It is straightforward to check that h j is strictly decreasing and log-convex for every j. 18 To see this, suppose that lim h /h = l > 0 the limit exists, since h is log-convex), where we have dropped the product subscript to ease notation. There exists x 0 > 0 such that h x)/hx) > l/2 for all x x 0. Integrating this inequality, we see that log hx) hx 0) ) > l 2 x x 0) for all x > x 0. Taking exponentials on both side, and letting x go to infinity, we obtain that lim h = 0. Conversely, lim h > 0 implies that lim h /h = 0. 10

11 then lim p Π f does not necessarily exist. For instance, with CES or MNL demands, firms profits do not have a limit when all prices go to infinity. 4 Equilibrium Analysis In this section, we provide an equilibrium analysis of the multiproduct-firm pricing game. In the first part, we adopt an aggregative games approach to prove existence of equilibrium. In the second part, we investigate how firm behavior is affected by changes in the aggregator. In the third part, we study the equilibrium properties, both from a positive and normative point of view, and comparative statics. In the fourth part, we discuss extensions and provide a cookbook for applied work. Finally, we provide conditions under which the equilibrium is unique. 4.1 An Aggregative Games Approach to Equilibrium Existence There are three main difficulties associated with the equilibrium existence problem. First, Π f is not necessarily quasi-concave in p j ) j f. 19 Second, Π f is not necessarily upper semicontinuous in p j ) j f. 20,21 Third, if f is a multiproduct firm, then Π f is neither supermodular nor log-supermodular in p j ) j f. 22 The first two difficulties imply that standard existence theorems for compact games such as Nash or Glicksberg s theorems) based on Kakutani s fixed-point theorem cannot be applied. The last two difficulties imply that existence theorems based on supermodularity theory and Tarski s fixed-point theorem see Milgrom and Roberts, 1990; Vives, 1990, 2000; Topkis, 1998) have no bite. The second and, to some extent, the third) difficulty prevents us from using Jensen 2010) s existence theorem for aggregative games with monotone best replies. The idea behind our existence proof is to reduce the dimensionality of the problem in two ways. First, we show that a firm s optimal price vector can be fully summarized by a unidimensional sufficient statistic, which is pinned down by a single equation in one unknown. Second, the pricing game is aggregative see Selten, 1970), in that the profit of a firm depends only on its own profile of prices and the uni-dimensional sufficient statistic = j N h jp j ). In the following, we present a semi-formal sketch of our existence proof. We refer the reader to Appendix A for details. In this sketch, we introduce the key concepts of ι-markup, pricing function, fitting-in function and aggregate fitting-in function, which will prove useful 19 Spady 1984) and anson and Martin 1996) provide examples of multiproduct-firm pricing games with MNL demand in which quasi-concavity fails. 20 To see this, suppose that demand is CES, and that firm f = {k} is a single-product firm. If firm f s rivals are setting infinite prices for all their products, then Π f = σ 1)p k c k )/p k for every p k > 0. It follows that Π f goes to σ 1 as p k goes to infinity. This is strictly greater than 0, which is the profit firm f receives when it sets p k =. Therefore, Π f is not upper semi-continuous in p k. 21 If profit functions had been defined over 0, ) N instead of 0, ] N, then payoff functions would be upper semi-continuous, but the lack of compactness would become an issue. 22 See Online Appendix II.2. 11

12 to describe the equilibria of our pricing game, and to understand our comparative statics results. Fix a pricing game h j ) j N, F, c j ) j N ). For the sake of expositional simplicity, suppose that first-order conditions are sufficient for optimality. Ignoring the possibility of infinite prices for the time being, the first-order conditions for each firm s profit maximization problem hold at price vector p R N ++ if and only if for every f F and k f, π f = h k p k) 1 p k c k h k p p k) k p k p k h k p k) + j f where = j N h h jp j ) is the aggregator. Let ι k p k ) = p k p is a pricing equilibrium if and only if p k c k ι k p k ) = 1 + p k j f and = j N h j p j ). ) p j c j ) h jp j ) = 0, 4) k p k) h k p k). Then, the profile of prices p j c j ) h jp j ), f F, k f, 5) We learn two facts from equation 5). First, for a given f F, the right-hand side of equation 5) is independent of the identity of k f. It follows that, in any Nash equilibrium, for any f F, and for all k, l f, p k c k ι k p k ) = p l c l ι l p l ). p k p l Put differently, there exists a scalar µ f, which we call firm f s ι-markup, such that p k c k p k ι k p k ) = µ f for every k f. We say that firm f s profile of prices, p k ) k f satisfies the common ι- markup property. Second, we see from equation 5) that µ f = 1 + Π f p). Put differently, firm f s equilibrium profit is equal to the value of its ι-markup minus one. 23 The constant ι-markup property can be interpreted as follows. Consider a hypothetical single-product firm selling product k. Suppose that this firm behaves in a monopolistically competitive way, in the sense that it does not internalize the impact of its price on aggregator. Firm k therefore faces demand h k p k)/ and, since it takes as given, believes that the price elasticity of demand for its product is equal to the elasticity of h k p k), which is precisely ι k p k ). Therefore, firm k prices according to the inverse elasticity rule: p k c k p k = 1 ι k p k ). In our model, firm f internalizes its impact on the aggregator level as well as self-cannibalization effects. It therefore prices in a less aggressive way, according to modified inverse elasticity rule p k c k p k = µf, with ι k p k ) µf > 1. Put differently, firm f sets the same price that a firm would set under monopolistic competition, if that firm believed that the price 23 Recall that market size has been normalized to unity. Without this normalization, firm f s equilibrium profit is µ f 1 times market size. 12

13 elasticity of demand is equal to ι k p k )/µ f, instead of ι k p k ). What is remarkable is that the ι-markup µ f, which summarizes the impact of firm f s behavior on, is firm-specific, rather than product-specific. Next, we use ι-markups to reduce the dimensionality of firms profit maximization problems. Note first that equation 5) can be rewritten as follows: For every f F, µ f = p j c j h j p j ) h jp j )) 2 p j p j h j f j p j) h j }{{}. j) }{{} } =ι j p j ) {{ } γ j p j ) =µ f Defining γ j h j) 2 /h j, and rearranging, we obtain: ) µ f 1 1 γ j p j ) = 1. 6) j f Suppose that function p k p k c k p k ι k p k ) is one-to-one for every k N, and denote its inverse function by r k ). We call r k the pricing function for product k. Then, using equation 6), firm f s pricing strategy can be fully described by a uni-dimensional variable, µ f, such that ) µ f 1 1 γ j r j µ f )) = 1. 7) j f Suppose that equation 7) has a unique solution in µ f, denoted m f ). We call m f firm f s fitting-in function. Then, the equilibrium existence problem boils down to finding an such that = h j rj m f ) )). f F j f }{{} Γ) In the parlance of aggregative games, Γ is the aggregate fitting-in function. The equilibrium existence problem reduces to finding a fixed point of that function. As we will see later on, the aggregative games approach is also useful to establish equilibrium uniqueness: The pricing game has a unique equilibrium if the following index condition is satisfied: Γ ) < 1 whenever Γ) =. This informal exposition leaves a number of questions open. Are first-order conditions sufficient for optimality? Can infinite prices be accommodated? Is function p k p k c k p k ι k p k ) one-to-one for every k? Are fitting-in functions well-defined? Does the aggregate fitting-in function have a fixed point? We need one assumption to answer all these questions in the affirmative. Assumption 1. For every j N and p j > 0, ι jp j ) 0 whenever ι j p j ) > 1. 13

14 Theorem 1. Suppose that the demand system h j ) j N satisfies Assumption 1. Then, the pricing game h j ) j N, F, c j ) j N ) has a pricing equilibrium for every F and c j ) j N. The set of equilibrium aggregator levels coincides with the set of fixed points of the aggregate fitting-in function Γ. If is an equilibrium aggregator level, then, in the associated equilibrium, consumer surplus is given by log, firm f F earns profit m f ) 1, and product k f is priced at r k m f )). Proof. See Appendix A. Broadly speaking, Assumption 1 says that for every product j, ι j, the price elasticity of the monopolistic competition demand for product j, should be non-decreasing in p j. This condition is sometimes called Marshall s second law of demand. It clearly holds with CES and MNL demands, where ι j p j ) is equal to σ and p j /λ, respectively. We discuss how it can be relaxed in Section 4.4. In the following, we denote by ι the set of C 3, strictly decreasing and log-convex functions from R ++ to R ++ such that Assumption 1 holds. 4.2 Properties of Fitting-in and Pricing Functions In this section, we study the properties of the product-level pricing function r k and the firmlevel fitting-in function m f, and discuss how these properties shape the behavior of firm f. 24 These functions turn out to be convenient for deriving and interpreting comparative statics in Section 4.3. For every product k N, denote µ k = lim ι k, and let p mc k be the unique solution of equation p k c k p k ι k p k ) = 1. Note that p mc k is the price at which product k would be sold under monopolistic competition. µ k is the highest ι-markup that product k can support. Proposition 2 Pricing function). r k is continuous and strictly increasing on 1, µ k ). Moreover, lim 1 r k = p mc k, lim µ k r k =, and r k µ f ) = for every µ f µ k. In words, the price of product k increases when ι-markup µ f increases. If µ f approaches unity the monopolistic competition ι-markup), then p k approaches p mc k the monopolistic competition price). If µ f is above µ k, then ι k p k )/µ f, the adjusted price elasticity of demand under monopolistic competition, is strictly lower than unity for every p k. Therefore, firm f sets an infinite price for product k, i.e., it does not supply product k. Next, we turn our attention to firm-level fitting-in function m f. For every firm f, let µ f = max j f µ j denote the highest ι-markup that firm f can sustain. Proposition 3 Fitting-in function). m f is continuous and strictly decreasing on 0, ). Moreover, lim 0 m f = µ f, and lim m f = 1. As competition intensifies increases), firm f reacts by lowering its ι-markup. As the industry approaches the monopolistic competition limit ), m f tends to 1, the ι- markup under monopolistic competition. Combining Propositions 2 and 3, we see that, as 24 These properties are rigorously established in Appendix A. 14

15 competition intensifies, firm f lowers the prices of all its products. An immediate consequence is that the aggregate fitting-in function Γ is strictly increasing. Propositions 2 and 3 also tell us how firm f s product range varies with the intensity of competition. To fix ideas, let f = {1, 2,..., N f }, and assume that products are ranked as follows: µ 1 > µ 2 >... > µ N f. When competition is very soft close to 0), m f ) is close to µ 1, and strictly higher than µ 2. Therefore, only product 1 is supplied. As increases, m f ) decreases, and eventually crosses µ 2, so that product 2 starts being supplied as well. When approaches the monopolistic competition limit, m f ) is strictly lower than µ N f, and firm f therefore sells all of its products. To summarize, the model predicts that a firm tends to sell more products when it operates in a more competitive environment. The intuition is easiest to grasp in the case where firm f sells CES products with heterogeneous σ s: For every j f, h j p j ) = a j p 1 σ j j. Then, ι j p j ) = σ j for every j, and σ 1 > σ 2 >... > σ N f. In the discrete/continuous choice interpretation of the demand system, the conditional demand for product j is given by d log h j p j )/dp j = σ j 1)/p j. The conditional profit made on product j is therefore σ j 1) p j c j p j. The supremum of that conditional profit is σ j 1. If competition is soft, then firm f has little to worry about consumers substituting away from its products. Firm f cares first and foremost about selling product 1, which is its most profitable product. It therefore sets a high price for product 1, earns a profit close to σ 1 1, and shuts down its other, less profitable, products, to prevent consumers from purchasing them. In other words, when competition is soft, firm f is mostly concerned about self-cannibalization effects, and therefore has an incentive to withdraw relatively unprofitable products. On the other hand, as competition intensifies, self-cannibalization effects become less important, and firm f worries more about consumers switching to other firms products. The firm therefore has an incentive to flood the market with its products so as to increase the probability that one of its products ends up being chosen by consumers. 4.3 Properties of Equilibria and Comparative Statics Markups. Our class of demand systems can generate rich patterns of equilibrium markups within a firm s product portfolio. To see this, let us first consider the special case of CES products with common σ i.e., h j p j ) = a j p 1 σ j for all j N ). In this case, ι j = σ for all j, and the common ι-markup property implies that, in equilibrium, p j c j p j = µf for all σ j f. Therefore, firm f sets the same Lerner index for all the products in its portfolio, and thus charges higher absolute markups on products that it produces less efficiently since p j c j = µf c σ µ f j is increasing in c j ). These markup patterns are not robust to changes in the demand system. Suppose for instance that all products are still CES products, but with potentially heterogeneous σ s i.e., h j p j ) = a j p 1 σ j j for all j N ). Then, in equilibrium, p j c j p j = µf σ j, and firm f no longer sets the same Lerner index over all its products unless all these products share the same σ). Similarly, it does not necessarily charge higher absolute markups on high marginal cost 15

16 products. The same point could be made about the other special case in which all products are MNL with common λ s h j p j ) = exp a j p j ) λ for all j N ). In this case, in equilibrium, a multiproduct firm charges the same absolute markup over all its products since ι j p j ) = p j /λ for all j), and sets a lower Lerner index on high marginal cost products. Again, this can be overturned by allowing the λ s to differ across products. 25 It is straightforward to construct fully parametric demand systems that give rise to a richer pattern of within-firm markups than either CES or MNL. 26 More generally, the pattern of markups within a firm s product portfolio depends on demand-side conditions, as captured by functions ι j ) j f, and on supply-side considerations c j ) j f ). Comparing equilibria. If we know that is an equilibrium aggregator level, then we can compute consumer surplus given by log ), the profit of firm f F given by m f ) 1) and the price of product k f given by r k m f )). Moreover, Propositions 2 and 3 imply that if there are multiple equilibria, then these equilibria can be Pareto-ranked among firms, with this ranking being the inverse of consumers ranking of equilibria: Proposition 4. Suppose that there are two pricing equilibria with aggregators 1 and 2 > 1, respectively. Then, each firm f F makes a strictly larger profit in the first equilibrium with aggregator 1), whereas consumers indirect utility is higher in the second equilibrium with aggregator 2). In addition, the set of equilibrium aggregator levels has a maximal and a minimal element. Proof. See Online Appendix IX.1. Welfare analysis. Next, we analyze the welfare distortions arising from multiproduct-firm oligopoly pricing. An immediate observation is that firms pricing is constrained efficient in the following sense: Firm f s equilibrium prices r k m f ))) k f maximize social welfare subject to the constraint that the firm s contribution to the aggregator, f, is held fixed at its equilibrium value. The reason is that consumer surplus and rivals profits are held fixed by the constraint, but firm f s prices maximize its profit by definition. 25 Björnerstedt and Verboven 2016) analyze a merger in the Swedish market for painkillers, and find that a CES demand specification or, in the authors own words, a constant expenditure demand specification) with random coefficients gives rise to more plausible markup predictions than an MNL demand specification with random coefficients. 26 Consider, for example, the following family of h functions: For every λ > 0, φ [0, 1] and p > 0, h φ,λ p) = { ) exp λ pφ 1+φ 2 φ if φ > 0, p λ if φ = 0. It is easy to check that h φ,λ converges pointwise to h 0,λ i.e., CES) when φ goes to zero, and to MNL when φ goes to 1, and that h φ,λ ι for every φ, λ. 16

17 As there are no within-firm pricing distortions, this leaves us with two types of distortions. The first distortion comes from the fact that, under oligopoly, firms set positive markups. This implies that, the equilibrium aggregator level, is strictly lower than the aggregator level under perfect competition j N h jc j )). The second distortion is due to the fact that, conditional on aggregator level, some firms are contributing too much to, while some others are contributing too little. This is easily seen by maximizing social welfare subject to the constraint that consumer surplus is equal to log : 27 max p p k c k ) k N h k p k) j N h jp j ) s.t. log j N h j p j ) = log. The first-order condition for product i can be written as follows: 28 p i c i ι i p i ) = 1 Λ + p k c k ) h k p k), 8) p i k N where Λ is the Lagrange multiplier associated with the consumer surplus constraint. Since the right-hand side of equation 8) does not depend on i, it follows that the profile of prices p k ) k N satisfies the common ι-markup property. Let µ be the optimal ι-markup. Then, the profile of N first-order conditions boils down to the following optimality condition: ) µ 1 1 γ j r j µ )) = 1 Λ. 9) j N This means that pricing equilibrium is constrained efficient if and only if m f ) = m g ) for every f, g F. This condition is unlikely to hold in general. 29 When it does not hold, some firms set their ι-markups above µ and end up producing too little, while some other firms set their ι-markups below µ and therefore produce too much. 30 Put differently, social welfare can be increased by raising f for some firm f while reducing g for some other firm g such that f + g remains constant. Whether a given firm contributes too much or too little to the aggregator can be assessed by comparing equations 7) and 9). Comparative statics. Although our pricing game is not supermodular, we can exploit its aggregative structure to perform comparative statics on the set of equilibria. The approach is similar to to the one in Corchon 1994) and Acemoglu and Jensen 2013), and can be 27 We drop the consumer surplus term in the social welfare expression, since the constraint implies that that term is a constant. 28 Infinite prices can be handled by defining generalized first-order conditions, as we do in Appendix A. 29 For instance, with CES or MNL demands, this condition holds if and only if all firms are of the same type. See Section If all firms set their ι-markups above µ resp. below µ ), then the consumer surplus constraint is violated. 17